write a quadratic function to model the graph to the right. f(x)=□

Explanation:

Step1: Identify vertex form of quadratic

The vertex - form of a quadratic function is $f(x)=a(x - h)^2 + k$, where $(h,k)$ is the vertex of the parabola. From the graph, the vertex is $(-1,2)$, so $h=-1$ and $k = 2$. Then $f(x)=a(x + 1)^2+2$.

Step2: Find the value of a

We can use another point on the graph, say $(0,5)$. Substitute $x = 0$ and $y = 5$ into $f(x)=a(x + 1)^2+2$. We get $5=a(0 + 1)^2+2$. Simplify the equation: $5=a+2$. Subtract 2 from both sides: $a=3$.

Step3: Write the quadratic function

Substitute $a = 3$ into $f(x)=a(x + 1)^2+2$. So $f(x)=3(x + 1)^2+2$. Expand it: $f(x)=3(x^{2}+2x + 1)+2=3x^{2}+6x+3 + 2=3x^{2}+6x + 5$.

Answer:

$3x^{2}+6x + 5$